Main:z from

Percentage Accurate: 91.4% → 99.6%

Time: 27.0s

Alternatives: 22

Speedup: 0.7×

Specification

Math

\[\begin{array}{l} \\ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
   (- (sqrt (+ z 1.0)) (sqrt z)))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 91.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
   (- (sqrt (+ z 1.0)) (sqrt z)))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{x + 1}\\ t_3 := {\left(y + 1\right)}^{0.25}\\ t_4 := \sqrt{t + 1}\\ t_5 := \sqrt{y + 1}\\ t_6 := \left(t\_2 + \sqrt{x}\right) \cdot \left(\sqrt{y} + t\_5\right)\\ \mathbf{if}\;t\_1 - \sqrt{z} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(t\_4 - \sqrt{t}\right) + \left(\mathsf{fma}\left(\frac{1}{t\_6}, t\_2 + t\_5, \frac{\sqrt{y}}{t\_6}\right) + \mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \frac{\sqrt{x}}{t\_6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_3, t\_3, \left(\left(1 - \sqrt{z}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + \left(\frac{1}{\sqrt{t} + t\_4} + t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (pow (+ y 1.0) 0.25))
        (t_4 (sqrt (+ t 1.0)))
        (t_5 (sqrt (+ y 1.0)))
        (t_6 (* (+ t_2 (sqrt x)) (+ (sqrt y) t_5))))
   (if (<= (- t_1 (sqrt z)) 5e-5)
     (+
      (- t_4 (sqrt t))
      (+
       (fma (/ 1.0 t_6) (+ t_2 t_5) (/ (sqrt y) t_6))
       (fma (sqrt (/ 1.0 z)) 0.5 (/ (sqrt x) t_6))))
     (fma
      t_3
      t_3
      (+
       (- (- 1.0 (sqrt z)) (+ (sqrt y) (sqrt x)))
       (+ (/ 1.0 (+ (sqrt t) t_4)) t_1))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double t_2 = sqrt((x + 1.0));
	double t_3 = pow((y + 1.0), 0.25);
	double t_4 = sqrt((t + 1.0));
	double t_5 = sqrt((y + 1.0));
	double t_6 = (t_2 + sqrt(x)) * (sqrt(y) + t_5);
	double tmp;
	if ((t_1 - sqrt(z)) <= 5e-5) {
		tmp = (t_4 - sqrt(t)) + (fma((1.0 / t_6), (t_2 + t_5), (sqrt(y) / t_6)) + fma(sqrt((1.0 / z)), 0.5, (sqrt(x) / t_6)));
	} else {
		tmp = fma(t_3, t_3, (((1.0 - sqrt(z)) - (sqrt(y) + sqrt(x))) + ((1.0 / (sqrt(t) + t_4)) + t_1)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = Float64(y + 1.0) ^ 0.25
	t_4 = sqrt(Float64(t + 1.0))
	t_5 = sqrt(Float64(y + 1.0))
	t_6 = Float64(Float64(t_2 + sqrt(x)) * Float64(sqrt(y) + t_5))
	tmp = 0.0
	if (Float64(t_1 - sqrt(z)) <= 5e-5)
		tmp = Float64(Float64(t_4 - sqrt(t)) + Float64(fma(Float64(1.0 / t_6), Float64(t_2 + t_5), Float64(sqrt(y) / t_6)) + fma(sqrt(Float64(1.0 / z)), 0.5, Float64(sqrt(x) / t_6))));
	else
		tmp = fma(t_3, t_3, Float64(Float64(Float64(1.0 - sqrt(z)) - Float64(sqrt(y) + sqrt(x))) + Float64(Float64(1.0 / Float64(sqrt(t) + t_4)) + t_1)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(y + 1.0), $MachinePrecision], 0.25], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$2 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[y], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], 5e-5], N[(N[(t$95$4 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 / t$95$6), $MachinePrecision] * N[(t$95$2 + t$95$5), $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] / t$95$6), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] * 0.5 + N[(N[Sqrt[x], $MachinePrecision] / t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * t$95$3 + N[(N[(N[(1.0 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[t], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 5.00000000000000024e-5

   1. Initial program 90.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}} + \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. flip-- N/A

         \[\leadsto \left(\left(\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}} + \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. frac-add N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right) + \left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right)}{\left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right) + \left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right)}{\left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 91.4%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(\left(y + 1\right) - y, \sqrt{x} + \sqrt{1 + x}, \left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\left(1 + x\right) - x\right)\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\frac{\color{blue}{1 + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{1 + y}\right)\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right)} + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \color{blue}{\left(\sqrt{1 + y} + \sqrt{y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \color{blue}{\left(\sqrt{1 + y} + \sqrt{y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\color{blue}{\sqrt{1 + y}} + \sqrt{y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{\color{blue}{y + 1}} + \sqrt{y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{\color{blue}{y + 1}} + \sqrt{y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 97.2

          \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y + 1} + \color{blue}{\sqrt{y}}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 97.2%

      \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y + 1} + \sqrt{y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \left(\sqrt{x} \cdot \frac{1}{\left(\sqrt{x} + \sqrt{1 + x}\right) \cdot \left(\sqrt{y} + \sqrt{1 + y}\right)} + \left(\sqrt{y} \cdot \frac{1}{\left(\sqrt{x} + \sqrt{1 + x}\right) \cdot \left(\sqrt{y} + \sqrt{1 + y}\right)} + \left(\sqrt{1 + x} \cdot \frac{1}{\left(\sqrt{x} + \sqrt{1 + x}\right) \cdot \left(\sqrt{y} + \sqrt{1 + y}\right)} + \sqrt{1 + y} \cdot \frac{1}{\left(\sqrt{x} + \sqrt{1 + x}\right) \cdot \left(\sqrt{y} + \sqrt{1 + y}\right)}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \sqrt{x} \cdot \frac{1}{\left(\sqrt{x} + \sqrt{1 + x}\right) \cdot \left(\sqrt{y} + \sqrt{1 + y}\right)}\right) + \left(\sqrt{y} \cdot \frac{1}{\left(\sqrt{x} + \sqrt{1 + x}\right) \cdot \left(\sqrt{y} + \sqrt{1 + y}\right)} + \left(\sqrt{1 + x} \cdot \frac{1}{\left(\sqrt{x} + \sqrt{1 + x}\right) \cdot \left(\sqrt{y} + \sqrt{1 + y}\right)} + \sqrt{1 + y} \cdot \frac{1}{\left(\sqrt{x} + \sqrt{1 + x}\right) \cdot \left(\sqrt{y} + \sqrt{1 + y}\right)}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \sqrt{x} \cdot \frac{1}{\left(\sqrt{x} + \sqrt{1 + x}\right) \cdot \left(\sqrt{y} + \sqrt{1 + y}\right)}\right) + \left(\sqrt{y} \cdot \frac{1}{\left(\sqrt{x} + \sqrt{1 + x}\right) \cdot \left(\sqrt{y} + \sqrt{1 + y}\right)} + \left(\sqrt{1 + x} \cdot \frac{1}{\left(\sqrt{x} + \sqrt{1 + x}\right) \cdot \left(\sqrt{y} + \sqrt{1 + y}\right)} + \sqrt{1 + y} \cdot \frac{1}{\left(\sqrt{x} + \sqrt{1 + x}\right) \cdot \left(\sqrt{y} + \sqrt{1 + y}\right)}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Applied rewrites 99.8%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \frac{\sqrt{x}}{\left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\right) + \mathsf{fma}\left(\frac{1}{\left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}, \sqrt{y + 1} + \sqrt{x + 1}, \frac{\sqrt{y}}{\left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 5.00000000000000024e-5 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   1. Initial program 97.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\left(\sqrt{t + 1} - \sqrt{t}\right)} \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\sqrt{t + 1}} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\sqrt{t + 1} \cdot \color{blue}{\sqrt{t + 1}} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\left(t + 1\right)} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \color{blue}{\sqrt{t}} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \sqrt{t} \cdot \color{blue}{\sqrt{t}}}{\sqrt{t + 1} + \sqrt{t}} \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \color{blue}{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\left(t + 1\right) - t}}{\sqrt{t + 1} + \sqrt{t}} \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - t}{\color{blue}{\sqrt{t} + \sqrt{t + 1}}} \]

      12. lower-+.f64 98.8

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - t}{\color{blue}{\sqrt{t} + \sqrt{t + 1}}} \]

   4. Applied rewrites 98.8%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   7. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\left(\left(\left(\frac{1}{\sqrt{t} + \sqrt{t + 1}} + \sqrt{z + 1}\right) + \sqrt{y + 1}\right) + 1\right) - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.2%

      \[\leadsto \mathsf{fma}\left({\left(y + 1\right)}^{0.25}, \color{blue}{{\left(y + 1\right)}^{0.25}}, \left(\frac{1}{\sqrt{t} + \sqrt{t + 1}} + \sqrt{z + 1}\right) + \left(\left(1 - \sqrt{z}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + z} - \sqrt{z} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\mathsf{fma}\left(\frac{1}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y} + \sqrt{y + 1}\right)}, \sqrt{x + 1} + \sqrt{y + 1}, \frac{\sqrt{y}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y} + \sqrt{y + 1}\right)}\right) + \mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \frac{\sqrt{x}}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{y} + \sqrt{y + 1}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(y + 1\right)}^{0.25}, {\left(y + 1\right)}^{0.25}, \left(\left(1 - \sqrt{z}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + \left(\frac{1}{\sqrt{t} + \sqrt{t + 1}} + \sqrt{1 + z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := t\_1 - \sqrt{z}\\ t_3 := \sqrt{y + 1}\\ t_4 := \sqrt{y} + \sqrt{x}\\ t_5 := \sqrt{t + 1}\\ t_6 := \sqrt{x + 1}\\ t_7 := \left(\left(t\_3 - \sqrt{y}\right) + \left(t\_6 - \sqrt{x}\right)\right) + t\_2\\ \mathbf{if}\;t\_7 \leq 0.99999998:\\ \;\;\;\;\left(\frac{1}{t\_6 + \sqrt{x}} + t\_2\right) + \left(t\_5 - \sqrt{t}\right)\\ \mathbf{elif}\;t\_7 \leq 2.9999999:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{z} + t\_1} + t\_3\right) - t\_4\right) + t\_6\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 + t\_1\right) + \frac{1}{\sqrt{t} + t\_5}\right) - \left(t\_4 + \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 z)))
        (t_2 (- t_1 (sqrt z)))
        (t_3 (sqrt (+ y 1.0)))
        (t_4 (+ (sqrt y) (sqrt x)))
        (t_5 (sqrt (+ t 1.0)))
        (t_6 (sqrt (+ x 1.0)))
        (t_7 (+ (+ (- t_3 (sqrt y)) (- t_6 (sqrt x))) t_2)))
   (if (<= t_7 0.99999998)
     (+ (+ (/ 1.0 (+ t_6 (sqrt x))) t_2) (- t_5 (sqrt t)))
     (if (<= t_7 2.9999999)
       (+ (- (+ (/ 1.0 (+ (sqrt z) t_1)) t_3) t_4) t_6)
       (- (+ (+ 2.0 t_1) (/ 1.0 (+ (sqrt t) t_5))) (+ t_4 (sqrt z)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double t_2 = t_1 - sqrt(z);
	double t_3 = sqrt((y + 1.0));
	double t_4 = sqrt(y) + sqrt(x);
	double t_5 = sqrt((t + 1.0));
	double t_6 = sqrt((x + 1.0));
	double t_7 = ((t_3 - sqrt(y)) + (t_6 - sqrt(x))) + t_2;
	double tmp;
	if (t_7 <= 0.99999998) {
		tmp = ((1.0 / (t_6 + sqrt(x))) + t_2) + (t_5 - sqrt(t));
	} else if (t_7 <= 2.9999999) {
		tmp = (((1.0 / (sqrt(z) + t_1)) + t_3) - t_4) + t_6;
	} else {
		tmp = ((2.0 + t_1) + (1.0 / (sqrt(t) + t_5))) - (t_4 + sqrt(z));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + z))
    t_2 = t_1 - sqrt(z)
    t_3 = sqrt((y + 1.0d0))
    t_4 = sqrt(y) + sqrt(x)
    t_5 = sqrt((t + 1.0d0))
    t_6 = sqrt((x + 1.0d0))
    t_7 = ((t_3 - sqrt(y)) + (t_6 - sqrt(x))) + t_2
    if (t_7 <= 0.99999998d0) then
        tmp = ((1.0d0 / (t_6 + sqrt(x))) + t_2) + (t_5 - sqrt(t))
    else if (t_7 <= 2.9999999d0) then
        tmp = (((1.0d0 / (sqrt(z) + t_1)) + t_3) - t_4) + t_6
    else
        tmp = ((2.0d0 + t_1) + (1.0d0 / (sqrt(t) + t_5))) - (t_4 + sqrt(z))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + z));
	double t_2 = t_1 - Math.sqrt(z);
	double t_3 = Math.sqrt((y + 1.0));
	double t_4 = Math.sqrt(y) + Math.sqrt(x);
	double t_5 = Math.sqrt((t + 1.0));
	double t_6 = Math.sqrt((x + 1.0));
	double t_7 = ((t_3 - Math.sqrt(y)) + (t_6 - Math.sqrt(x))) + t_2;
	double tmp;
	if (t_7 <= 0.99999998) {
		tmp = ((1.0 / (t_6 + Math.sqrt(x))) + t_2) + (t_5 - Math.sqrt(t));
	} else if (t_7 <= 2.9999999) {
		tmp = (((1.0 / (Math.sqrt(z) + t_1)) + t_3) - t_4) + t_6;
	} else {
		tmp = ((2.0 + t_1) + (1.0 / (Math.sqrt(t) + t_5))) - (t_4 + Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + z))
	t_2 = t_1 - math.sqrt(z)
	t_3 = math.sqrt((y + 1.0))
	t_4 = math.sqrt(y) + math.sqrt(x)
	t_5 = math.sqrt((t + 1.0))
	t_6 = math.sqrt((x + 1.0))
	t_7 = ((t_3 - math.sqrt(y)) + (t_6 - math.sqrt(x))) + t_2
	tmp = 0
	if t_7 <= 0.99999998:
		tmp = ((1.0 / (t_6 + math.sqrt(x))) + t_2) + (t_5 - math.sqrt(t))
	elif t_7 <= 2.9999999:
		tmp = (((1.0 / (math.sqrt(z) + t_1)) + t_3) - t_4) + t_6
	else:
		tmp = ((2.0 + t_1) + (1.0 / (math.sqrt(t) + t_5))) - (t_4 + math.sqrt(z))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = Float64(t_1 - sqrt(z))
	t_3 = sqrt(Float64(y + 1.0))
	t_4 = Float64(sqrt(y) + sqrt(x))
	t_5 = sqrt(Float64(t + 1.0))
	t_6 = sqrt(Float64(x + 1.0))
	t_7 = Float64(Float64(Float64(t_3 - sqrt(y)) + Float64(t_6 - sqrt(x))) + t_2)
	tmp = 0.0
	if (t_7 <= 0.99999998)
		tmp = Float64(Float64(Float64(1.0 / Float64(t_6 + sqrt(x))) + t_2) + Float64(t_5 - sqrt(t)));
	elseif (t_7 <= 2.9999999)
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(z) + t_1)) + t_3) - t_4) + t_6);
	else
		tmp = Float64(Float64(Float64(2.0 + t_1) + Float64(1.0 / Float64(sqrt(t) + t_5))) - Float64(t_4 + sqrt(z)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + z));
	t_2 = t_1 - sqrt(z);
	t_3 = sqrt((y + 1.0));
	t_4 = sqrt(y) + sqrt(x);
	t_5 = sqrt((t + 1.0));
	t_6 = sqrt((x + 1.0));
	t_7 = ((t_3 - sqrt(y)) + (t_6 - sqrt(x))) + t_2;
	tmp = 0.0;
	if (t_7 <= 0.99999998)
		tmp = ((1.0 / (t_6 + sqrt(x))) + t_2) + (t_5 - sqrt(t));
	elseif (t_7 <= 2.9999999)
		tmp = (((1.0 / (sqrt(z) + t_1)) + t_3) - t_4) + t_6;
	else
		tmp = ((2.0 + t_1) + (1.0 / (sqrt(t) + t_5))) - (t_4 + sqrt(z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[(N[(N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$6 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]}, If[LessEqual[t$95$7, 0.99999998], N[(N[(N[(1.0 / N[(t$95$6 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + N[(t$95$5 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, 2.9999999], N[(N[(N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] - t$95$4), $MachinePrecision] + t$95$6), $MachinePrecision], N[(N[(N[(2.0 + t$95$1), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[t], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$4 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.999999980000000011

   1. Initial program 25.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}} + \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. flip-- N/A

         \[\leadsto \left(\left(\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}} + \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. frac-add N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right) + \left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right)}{\left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right) + \left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right)}{\left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 32.0%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(\left(y + 1\right) - y, \sqrt{x} + \sqrt{1 + x}, \left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\left(1 + x\right) - x\right)\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\frac{\color{blue}{1 + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{1 + y}\right)\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right)} + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \color{blue}{\left(\sqrt{1 + y} + \sqrt{y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \color{blue}{\left(\sqrt{1 + y} + \sqrt{y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\color{blue}{\sqrt{1 + y}} + \sqrt{y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{\color{blue}{y + 1}} + \sqrt{y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{\color{blue}{y + 1}} + \sqrt{y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 85.2

          \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y + 1} + \color{blue}{\sqrt{y}}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 85.2%

      \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y + 1} + \sqrt{y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + x}} + \sqrt{x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 84.0

         \[\leadsto \left(\frac{1}{\sqrt{x + 1} + \color{blue}{\sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Applied rewrites 84.0%

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.999999980000000011 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.99999990000000016

   1. Initial program 96.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\sqrt{z + 1}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{z + 1} \cdot \color{blue}{\sqrt{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 96.8

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 96.8%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{x + 1}} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{x + 1}} + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   7. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(\frac{1}{\sqrt{z} + \sqrt{z + 1}} + \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   if 2.99999990000000016 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 98.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\left(\sqrt{t + 1} - \sqrt{t}\right)} \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\sqrt{t + 1}} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\sqrt{t + 1} \cdot \color{blue}{\sqrt{t + 1}} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\left(t + 1\right)} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \color{blue}{\sqrt{t}} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \sqrt{t} \cdot \color{blue}{\sqrt{t}}}{\sqrt{t + 1} + \sqrt{t}} \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \color{blue}{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\left(t + 1\right) - t}}{\sqrt{t + 1} + \sqrt{t}} \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - t}{\color{blue}{\sqrt{t} + \sqrt{t + 1}}} \]

      12. lower-+.f64 99.3

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - t}{\color{blue}{\sqrt{t} + \sqrt{t + 1}}} \]

   4. Applied rewrites 99.3%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   7. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(\left(\left(\frac{1}{\sqrt{t} + \sqrt{t + 1}} + \sqrt{z + 1}\right) + \sqrt{y + 1}\right) + 1\right) - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \left(2 + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} + \sqrt{z}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

       \[\leadsto \left(\left(2 + \sqrt{z + 1}\right) + \frac{1}{\sqrt{t} + \sqrt{t + 1}}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} + \sqrt{z}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 0.99999998:\\ \;\;\;\;\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 2.9999999:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + \sqrt{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 + \sqrt{1 + z}\right) + \frac{1}{\sqrt{t} + \sqrt{t + 1}}\right) - \left(\left(\sqrt{y} + \sqrt{x}\right) + \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+
    (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))
    (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y))))
   (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z))))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))) + (1.0d0 / (sqrt((y + 1.0d0)) + sqrt(y)))) + (1.0d0 / (sqrt((z + 1.0d0)) + sqrt(z)))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x))) + (1.0 / (Math.sqrt((y + 1.0)) + Math.sqrt(y)))) + (1.0 / (Math.sqrt((z + 1.0)) + Math.sqrt(z)))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))) + (1.0 / (math.sqrt((y + 1.0)) + math.sqrt(y)))) + (1.0 / (math.sqrt((z + 1.0)) + math.sqrt(z)))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x))) + Float64(1.0 / Float64(sqrt(Float64(y + 1.0)) + sqrt(y)))) + Float64(1.0 / Float64(sqrt(Float64(z + 1.0)) + sqrt(z)))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024230 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

  :alt
  (! :herbie-platform default (+ (+ (+ (/ 1 (+ (sqrt (+ x 1)) (sqrt x))) (/ 1 (+ (sqrt (+ y 1)) (sqrt y)))) (/ 1 (+ (sqrt (+ z 1)) (sqrt z)))) (- (sqrt (+ t 1)) (sqrt t))))

  (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) (- (sqrt (+ z 1.0)) (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t))))